MATH 115B Exam 1 Review
Exam 1 – Monday, October 5th
1. The marginal revenue and cost functions for a good are MR(q) = 500 - 0.4q and
MC(q)= 250 respectively.
(a) Is R(q) increasing or decreasing at q = 600 units? Explain.
(b) Is C(q) increasing or decreasing at q = 1000 units? Explain.
(c) Is P(q) increasing or decreasing at q = 800 units? Explain.
(d) How many units should be produced and sold in order to maximize revenue?
(e) How many units should be produced and sold in order to maximize profit?
2. Let .
(a) Use a difference quotient, with an increment of h=0.001 to estimate f′(0).
(b) Find the equation of the line that is tangent to the graph of at
x = 0.
3. The demand function for a good is given by, the fixed costs are $100,000, and the variable costs are given by .
(a) How many units could be sold at a price of $625.00?
(b) Give a formula for the revenue function for the good.
(c) What revenue would result from the sale of 800 units?
(d) Give a formula for the total cost function, C(q).
(e) How many units of the good can be produced for a total cost of $250,000?
4. Find the equation of the tangent line to the graph of at .
5. Data on the test markets and costs for a good are given below. All monetary amounts are in dollars and all quantities are single units.
Potential National Market: 5,000Test Markets
Market Number / Market Size / Price / Projected Yearly Sales
1 / 100 / $59.95 / 50
2 / 300 / $69.95 / 125
3 / 200 / $89.95 / 25
4 / 500 / $79.95 / 150
5 / 400 / $49.95 / 225
Cost Data
Fixed Cost: $100,000
Variable Costs
Quantity / Cost per unit
First 1000 units / $15
Next 500 units / $12
Further / $10
(a) The equation for the polynomial trend line that has been fitted to the data in the five test markets is given by D(q )=-0.000005q2 - 0.0011q + 92.54. Determine
the price per unit if 1900 units are sold.
(b) How much revenue would be earned if 1,900 units are produced and sold?
(c) What would be the total cost of producing 1,900 units?
(d) How much profit would be earned if 1,900 units are produced and sold
(e) Find the marginal profit at a production level of 1900 units.
6. Graphs of the revenue and cost functions for a product are shown in the following plot.
a. Estimate the smallest number of units at which the company would earn a positive profit.
b. Estimate the profit which would result from a production level which maximizes revenue.
c. Estimate the number of units which should be produced in order to maximize profit.
d. Estimate the company’s maximum possible profit.
7. A plot of Mega Bucks demand function is shown below. Suppose that the company sells 40 units at the demand function price. All monetary amounts are in dollars and all quantities are in numbers of units. Note that each grid box in the plot has an area of $20.
a. Shade the region in the plot that corresponds to the revenue from the sales.
b. Estimate the revenue that would result from the sale.
c. Estimate the consumer surplus that would result from the sale.
8. Let .
Compute the midpoint sum S3(f,[-4,2]).
9. Sketch the region whose area is represented by the definite integral